Two circle touch each other internally. Show that the tangent drawn to the two circle from any point on the common tangent, are equal in length. 

$PQ$ and $PT$ are tangents to a circle with centre $O$ and radius $5 \mathrm{cm}$. If $PQ=12 \mathrm{cm}$, then prove that the perimeter of the quadrilateral $PQOT$ is $34 \mathrm{cm}$.

In the following figure, $△ABC$ is circumscribing a circle, the length of $BC=$_____.

In the adjoining figure, $PS$ and $PT$ are tangents to the circle drawn from an external point $P$. $CD$ is a third tangent touching the circle at $Q$. If $PT=10 \mathrm{cm}$ and $CQ=2 \mathrm{cm}$, find the perimeter of $∆PCD$.

In the given figure,

find $PT$ in $\mathrm{cm}$, if $OP=41 \mathrm{cm}$ and $O{T}^{\text{'}}=9 \mathrm{cm}$.

In the given figure, $PQ$ is a chord of length $8 cm$ of a circle of radius $5 cm$ and centre $O$. The tangents at $P$ and $Q$ intersect at point $T$. Find the length of $TP$.

The radius of a circle with centre $O$ is $5 cm$. $P$ is a point at a distance $13 cm$ from $O$. $PQ$ and $PR$ are two tangents to this circle. Find the area of the quadrilateral $PQOR$.

In the figure, Two circles with centres $P$ and $Q$ touch externally at $R$. Tangents $AT$ and $BT$ meet the common tangent $TR$ at $T$. If $AP=6 cm$ and $PT=10 cm$, then $BT=$?

In the given figure a quadrilateral $\mathrm{ABCD}$ is drawn to circumscribe a circle. Prove that: $\mathrm{AB}+\mathrm{CD}=\mathrm{AD}+\mathrm{BC}$.

If all the sides of a parallelogram touch a circle, show that the parallelogram is a rhombus.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

In the figure, two tangents $\mathrm{TP} \text{and} \mathrm{TQ}$ are drawn to a circle with centre $\mathrm{O}$ from an external point $\mathrm{T}$. Prove that $\angle \mathrm{PTQ}=2\angle \mathrm{OPQ}$.